Question: The lifespans of snakes in a particular zoo are normally distributed. The average snake lives $22$ years; the standard deviation is $4.6$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a snake living longer than $12.8$ years.
$22$ $17.4$ $26.6$ $12.8$ $31.2$ $8.2$ $35.8$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $22$ years. We know the standard deviation is $4.6$ years, so one standard deviation below the mean is $17.4$ years and one standard deviation above the mean is $26.6$ years. Two standard deviations below the mean is $12.8$ years and two standard deviations above the mean is $31.2$ years. Three standard deviations below the mean is $8.2$ years and three standard deviations above the mean is $35.8$ years. We are interested in the probability of a snake living longer than $12.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the snakes will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the snakes will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $12.8$ years and the other half $({2.5\%})$ will live longer than $31.2$ years. The probability of a particular snake living longer than $12.8$ years is ${95\%} + {2.5\%}$, or $97.5\%$.